Non-Singular Matrix — Definition, Properties, and Examples
Non-Singular Matrix — Definition, Properties, and Examples
What Is a Non-Singular Matrix?
A non-singular matrix is a square matrix AAA whose determinant is not equal to zero, that is, (\det A \neq 0). Because its determinant is non-zero, a non-singular matrix always has an inverse, which is why it is also called an invertible matrix. Only square matrices (order (n \times n)) can be singular or non-singular, since only square matrices have a determinant.
Quick Reference:
Definition: A square matrix AAA with (\det A \neq 0).
Also called: invertible matrix
Condition: (\det A \neq 0) (equivalently, rank = n; rows/columns linearly independent)
Type: property of a square matrix
Used in: solving linear systems, matrix inverses, computer graphics, machine learning
The two tests below always agree, so you can use whichever is easier for the matrix in front of you:
- Determinant test: (\det A \neq 0 \Rightarrow ) non-singular.
- Rank test: a matrix of order nnn is non-singular if and only if its rank equals nnn, so every row (and column) is linearly independent.
Non-Singular vs. Singular Matrix
The cleanest way to hold the definition is by its opposite. A singular matrix has (\det A = 0); it is not invertible, and its rows or columns are linearly dependent.
| Feature | Non-singular matrix | Singular matrix |
|---|---|---|
| Determinant | (\det A \neq 0) | (\det A = 0) |
| Inverse | exists (invertible) | does not exist |
| Rank (order n) | rank = n | rank < n |
| Rows / columns | linearly independent | linearly dependent |
| System AX=B | unique solution | no unique solution |
How to Check If a Matrix Is Non-Singular
The procedure is short:
- Confirm the matrix is square (equal number of rows and columns). A non-square matrix is neither singular nor non-singular.
- Compute the determinant.
- If the determinant is non-zero, the matrix is non-singular; if it is zero, the matrix is singular.
Properties of Non-Singular Matrices
- The product of two non-singular matrices of the same order is non-singular: (\det(AB) = \det(A)\det(B)), and neither factor is zero.
- If AAA is non-singular, so is any non-zero scalar multiple (kA) (for (k \neq 0)).
- The inverse (A^{-1}) of a non-singular matrix is itself non-singular.
- The transpose (A^{T}) of a non-singular matrix is non-singular, since (\det A^{T} = \det A).
- The identity matrix is non-singular, with (\det I = 1).
Examples of Non-Singular Matrix
Example 1
Is A=[1 -4 \ 3 5] non-singular?
Apply (\det A = ad - bc): [ det A = (1)(5) - (-4)(3) = 5 + 12 = 17 ] Since (17 \neq 0), A is non-singular (invertible). Final answer: Non-singular, (\det A = 17).
Example 2
Is B=[3 6 \ 2 4] non-singular?
Compute the determinant:
[
det B = (3)(4) - (6)(2) = 12 - 12 = 0
]
The determinant is zero, so B is singular.
Final answer: Singular, (\det B = 0).
Example 3
Is C=[2 0 \ 0 7] non-singular?
[
det C = (2)(7) = 14
]
Since (14 \neq 0), C is non-singular.
Final answer: Non-singular, (\det C = 14).
Example 4
Is D=[4 -1 0 \ 2 3 5 \ -1 7 2] non-singular?
Expand along the first row:
[
det D = 4\begin{vmatrix} 3 & 5 \ 7 & 2 \end{vmatrix} + 1\begin{vmatrix} 2 & 5 \ -1 & 2 \end{vmatrix}
]
Compute each minor:
[
\begin{vmatrix} 3 & 5 \ 7 & 2 \end{vmatrix} = (3)(2) - (5)(7) = 6 - 35 = -29
]
[
\begin{vmatrix} 2 & 5 \ -1 & 2 \end{vmatrix} = (2)(2) - (5)(-1) = 4 + 5 = 9
]
Substitute:
[
det D = 4(-29) + 1(9) = -116 + 9 = -107
]
Since (-107 \neq 0), D is non-singular.
Final answer: Non-singular, (\det D = -107).
Example 5
Find k so that E=[k 2 \ 3 6] is singular.
Set the determinant to zero:
[
det E = 6k - 6 = 0 \Rightarrow k = 1
]
Final answer: Singular at (k = 1); non-singular for all (k \neq 1).
Example 6
If A and B are non-singular with (\det A = 5) and (\det B = -2), is AB non-singular? [ det(AB) = \det(A)\det(B) = (5)(-2) = -10 ] Since (-10 \neq 0), AB is non-singular. Final answer: Non-singular, (\det(AB) = -10).
Why the Non-Singular Property Matters: "Can this be undone?"
- Solving equations: A linear system (AX=B) has a unique solution (X=A^{-1}B) exactly when A is non-singular.
- Computer graphics: rotation and scaling matrices must be non-singular.
- Data and machine learning: many algorithms invert a matrix; a singular (or nearly singular) matrix breaks the computation.
What Are the Most Common Mistakes With Non-Singular Matrices?
Mistake 1: Judging singularity without computing the determinant
The correct way: Always compute (\det A = ad - bc).
Mistake 2: Testing a non-square matrix
The correct way: Only square matrices can be singular or non-singular.
Mistake 3: Confusing zero entries with a zero determinant
The correct way: A matrix full of zeros can still have a non-zero determinant.
Conclusion
- A non-singular matrix is a square matrix with (\det A \neq 0), and it is always invertible.
- Its opposite, the singular matrix, has (\det A = 0) and no inverse.
- Test by computing the determinant (or checking that rank = n).
- Products, transposes, and non-zero scalar multiples of non-singular matrices stay non-singular.